Make the radicand (the number inside the radical) of an even-numbered root negative
Make the argument of any logarithm zero or negative
Create other "no-no's" like
In your equation you have a denominator and you have a logarithm function. So you need to avoid the first and third situations described above.
The only denominator is "x" so to avoid a zero denominator we must exclude zero from the domain.
The argument of the log function is . We must avoid this becoming zero or negative. We'll deal with theses possibilities separately.
First we'll make sure the argument cannot be zero. If you understand fractions you know that the only way for a fraction to be equal to zero is if the numerator is zero. (Think about it.). So to keep the argument from being zero we need to avoid . We can solve this to find the x's we must exclude. This equation will factor:
The only way for this product to be zero is if one of the factors is zero: or
This gives or
At this point we know we must exclude -2, 0 and 2 from the domain.
The last part is the hardest. We must ensure that the argument to the log function is never negative. If we think about fractions and how they become negative we should realize that the only way is if the numerator and denominator have opposite signs: +/- or -/+. To figure out x-values we need to exclude we write this logic and solve:
For +/-: and
The right side inequality is done. We have some work to solve the left side inequality. First we'll factor it again: and
Now we have to figure out how the products become positive. We should quickly determine that both factors must have the same sign: + times + or - times -. The easiest way to figure out when both factors are positive is to realize that 1) (x-2) will always be smaller than (x+2); and 2) if the smaller factor is positive both factors would be positive. (Think about both of these.) So both factors will be positive if (x-2) > 0 or x > 2. So we have and . But how can a number be greater than 2 and less than 0 at the same time? Answer: it can't. So it is impossible to get a +/- fraction.
For -/+: and
Again the right side inequality is done. To solve the left side we will proceed as before. First factor: and
The only way for a product to be negative is if the two factors are of opposite signs: + times - or - times +. Since (x-2) is the smaller factor it must be the negative one and, consequently, (x+2) must be the positive one.
So now we have: and and
Which leads to: and and
The only numbers which fit all three inequalities are the numbers between 0 and 2. (Think about it.)
We finally have our domain! The domain is all Real numbersexcept: -2, 0, 2 and all the numbers between 0 and 2.
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